Question

The perpendicular distance from the origin to the focal chord drawn through the point \( (4, 5) \) to the parabola \( y^2 - 4y - 3x + 7 = 0 \) is:

• A. \( \frac{2}{5} \)
• B. \( \frac{1}{\sqrt{2}} \)
• C. \( \frac{1}{5} \)
• D. \( 1 \)

Hint:

When dealing with parabolas and focal chords, use the properties of the parabola’s equation and the geometry of the chord to calculate distances.

Answer 1

The Correct Option is C

The given parabola is \( y^2 - 4y - 3x + 7 = 0 \). First, complete the square to rewrite the equation in a standard form: \[ y^2 - 4y = 3x - 7 \] Complete the square for \( y \): \[ (y - 2)^2 = 3(x - \frac{7}{3}) \] This is a standard parabola equation with vertex \( (\frac{7}{3}, 2) \). Step 1: The point \( (4, 5) \) lies on the parabola. Using the properties of the focal chord, the equation for the distance from the origin to the focal chord is derived. Step 2: After performing the calculation, the perpendicular distance from the origin to the focal chord is \( \frac{1}{5} \). % Final Answer The perpendicular distance is \( \frac{1}{5} \).

Answer 2

Step 1: Convert the given equation into standard form
The given parabola is: \( y^2 - 4y - 3x + 7 = 0 \)
We aim to reduce it to the standard form by completing the square in \( y \):

Group the terms in \( y \):
\( y^2 - 4y = 3x - 7 \)

Complete the square:
\( y^2 - 4y + 4 = 3x - 7 + 4 \)
\( (y - 2)^2 = 3x - 3 \)

Thus, the equation becomes:
\( (y - 2)^2 = 3(x - 1) \)

Step 2: Identify key parameters of the parabola
The standard form of a rightward opening parabola is:
\( (y - k)^2 = 4a(x - h) \)
Here, \( h = 1 \), \( k = 2 \), and \( 4a = 3 \Rightarrow a = \frac{3}{4} \)
So, the focus of the parabola is at \( (h + a, k) = \left(1 + \frac{3}{4}, 2\right) = \left(\frac{7}{4}, 2\right) \)

Step 3: Use the concept of focal chord
A focal chord passes through the focus and intersects the parabola at two points. One of the points is given as \( (4, 5) \).

So, the chord passes through the focus \( \left(\frac{7}{4}, 2\right) \) and the point \( (4, 5) \).
We find the equation of the line through these two points:

Slope \( m = \frac{5 - 2}{4 - \frac{7}{4}} = \frac{3}{\frac{9}{4}} = \frac{12}{9} = \frac{4}{3} \)

Using point-slope form with point \( (4, 5) \):
\( y - 5 = \frac{4}{3}(x - 4) \)
Multiply through by 3:
\( 3y - 15 = 4x - 16 \Rightarrow 4x - 3y = 1 \)

Step 4: Find perpendicular distance from origin to this line
The perpendicular distance from origin \( (0, 0) \) to the line \( 4x - 3y = 1 \) is given by:
\( D = \frac{|4(0) - 3(0) - 1|}{\sqrt{4^2 + (-3)^2}} = \frac{1}{\sqrt{16 + 9}} = \frac{1}{\sqrt{25}} = \frac{1}{5} \)

Final Answer:
\( \frac{1}{5} \)